4z^2-4z-21=0

Solution for 4z^2-4z-21=0 equation:

Simplifying
4z2 + -4z + -21 = 0

Reorder the terms:
-21 + -4z + 4z2 = 0

Solving
-21 + -4z + 4z2 = 0

Solving for variable 'z'.

Begin completing the square.  Divide all terms by
4 the coefficient of the squared term: 

Divide each side by '4'.
-5.25 + -1z + z2 = 0

Move the constant term to the right:

Add '5.25' to each side of the equation.
-5.25 + -1z + 5.25 + z2 = 0 + 5.25

Reorder the terms:
-5.25 + 5.25 + -1z + z2 = 0 + 5.25

Combine like terms: -5.25 + 5.25 = 0.00
0.00 + -1z + z2 = 0 + 5.25
-1z + z2 = 0 + 5.25

Combine like terms: 0 + 5.25 = 5.25
-1z + z2 = 5.25

The z term is -1z.  Take half its coefficient (-0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
-1z + 0.25 + z2 = 5.25 + 0.25

Reorder the terms:
0.25 + -1z + z2 = 5.25 + 0.25

Combine like terms: 5.25 + 0.25 = 5.5
0.25 + -1z + z2 = 5.5

Factor a perfect square on the left side:
(z + -0.5)(z + -0.5) = 5.5

Calculate the square root of the right side: 2.34520788

Break this problem into two subproblems by setting 
(z + -0.5) equal to 2.34520788 and -2.34520788.

Subproblem 1
z + -0.5 = 2.34520788

Simplifying
z + -0.5 = 2.34520788

Reorder the terms:
-0.5 + z = 2.34520788

Solving
-0.5 + z = 2.34520788

Solving for variable 'z'.

Move all terms containing z to the left, all other terms to the right.

Add '0.5' to each side of the equation.
-0.5 + 0.5 + z = 2.34520788 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + z = 2.34520788 + 0.5
z = 2.34520788 + 0.5

Combine like terms: 2.34520788 + 0.5 = 2.84520788
z = 2.84520788

Simplifying
z = 2.84520788

Subproblem 2
z + -0.5 = -2.34520788

Simplifying
z + -0.5 = -2.34520788

Reorder the terms:
-0.5 + z = -2.34520788

Solving
-0.5 + z = -2.34520788

Solving for variable 'z'.

Move all terms containing z to the left, all other terms to the right.

Add '0.5' to each side of the equation.
-0.5 + 0.5 + z = -2.34520788 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + z = -2.34520788 + 0.5
z = -2.34520788 + 0.5

Combine like terms: -2.34520788 + 0.5 = -1.84520788
z = -1.84520788

Simplifying
z = -1.84520788

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {2.84520788, -1.84520788}